Here we find the most important use of primes: they are the unique building blocks of the multiplicative group of integers. In discussion of warfare you often hear the phrase "divide and conquer.
Many of the properties of an integer can be traced back to the properties of its prime divisors, allowing us to divide the problem literally into smaller problems. That is, divisibility by one fails to provide us any information about a.
Don't go feeling sorry for one, it is part of an important class of numbers call the units or divisors of unity. These are the elements numbers which have a multiplicative inverse. In some number systems there are infinitely many units.
So indeed there was a time that many folks defined one to be a prime, but it is the importance of units in modern mathematics that causes us to be much more careful with the number one and with primes.
See also the technical note in The prime Glossary ' definition. Numbers like 10 and 36 and 49 that can be composed as products of smaller counting numbers are called composite numbers. Numbers like this are called prime numbers. That captures the idea well, but is not a good enough definition, because it has too many loopholes. A prime number is a positive integer that has exactly two distinct whole number factors or divisors , namely 1 and the number itself.
Well, the definition rules it out. Even the informal idea rules it out: it cannot be built by multiplying other whole numbers. But why rule it out?! Why not include it?
Mathematics is not arbitrary. Well, if we include 1, there are infinitely many ways to write 12 as a product of primes. In fact, if we call 1 a prime, then there are infinitely many ways to write any number as a product of primes. Including 1 trivializes the question. Excluding it leaves only these cases:.
But there are similar number sets that have an infinite number of units. As sets like this became objects of study, it makes sense that the definitions of unit, irreducible, and prime would need to be carefully delineated.
In particular, if there are number sets with an infinite number of units, it gets more difficult to figure out what we mean by unique factorization of numbers unless we clarify that units cannot be prime. While I am not a math historian or a number theorist and would love to read more about exactly how this process took place before speculating further, I think this is one development Caldwell and Xiong allude to that motivated the exclusion of 1 from the primes.
As happens so often, my initial neat and tidy answer for why things are the way they are ended up being only part of the story. Thanks to my friend for asking the question and helping me learn more about the messy history of primality. The views expressed are those of the author s and are not necessarily those of Scientific American. Follow Evelyn Lamb on Twitter. Already a subscriber?
Sign in. Thanks for reading Scientific American. Create your free account or Sign in to continue. See Subscription Options. Discover World-Changing Science. In the positive whole numbers, each prime number p has two properties: The number p cannot be written as the product of two whole numbers, neither of which is a unit.
Get smart. This is another topic you might not have thought about since middle school. The Fundamental Theorem of Arithmetic states that every composite number can be broken down into a unique product of prime numbers. Prime factorization is the process that finds the prime-number products of a given composite number. Prime factorizations are often represented with exponents.
In a Factor-T, you will have two columns: the left column with the prime numbers and the right column with progressively smaller quotients from division. Step 1: Determine a prime number that divides into your given number. Write that number in the left column. Step 3: Now look for a prime number that divides into the quotient, 39 in this case. Continue steps 1 and 2 until you have a one in the right-hand column. The numbers on the left are your prime factorization!
The prime factorization of 78 is 2 x 3 x No other number has this prime factorization.
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